> [!NOTE] Definition (Eigenpair of linear operator) > Consider a [[Linear maps|linear operator]] $\varphi:V \to V.$ We say that $\lambda\in\mathbb{R}$ is an **eigenvalue** of $\varphi$ if there exists some nonzero $v\in V$ such that $\varphi(v)=\lambda v$We call $v$ an **eigenvector** of $\varphi$ corresponding to eigenvalue $\lambda.$ >[!Note] Definition (Eigenpair of square matrix) >Let $A\in \mathbb{R}^{n\times n}$ be a [[Real Square Matrices|square matrix]]. We say that $\lambda\in \mathbb{R}$ is an **eigenvalue** of $A$ if there exists some nonzero $\underline{v} \in \mathbb{R}^{n}$ such that $A\underline{v}=\lambda \underline{v}$We call $\underline{v}$ an **eigenvector** of $A$ corresponding to the eigenvalue $\lambda.$ > [!Example] Example (Eigenpairss in $\mathbb{R}^{2\times{2}}$) > The *eigenvalues* $\lambda$ of a $A \in M_{2\times 2} (\mathbb{R})$ are given as the roots to its characteristic polynomial. , They satisfy the equation $\begin{align} 0 &= \det(\lambda I - A) \\ &= \det \begin{pmatrix} \lambda - a_{1,1} & -a_{1,2} \\ -a_{2,1} & \lambda - a_{2,2} \end{pmatrix} \\ &= (\lambda-a_{1,1})(\lambda - a_{2,2}) - a_{2,1} a_{1,2} \\ &= \lambda^{2} - (a_{1,1} +a_{2,2}) \lambda + a_{1,1} a_{2,2} - a_{2,1} a_{1,2} \\ &= \lambda^{2}- \mathrm{Tr}(A) \lambda + \det (A) \end{align}$ # Properties > [!NOTE] Definition (Characteristic Polynomial) > Contents > [!NOTE] Definition (Eigenspace) > Contents # Application - [[Diagonalisation of Square Matrix]].